Large Rotation Rates Research Articles

Flow past a rotating cylinder

Flow past a spinning circular cylinder placed in a uniform stream is investigated via two-dimensional computations. A stabilized finite element method is utilized to solve the incompressible Navier–Stokes equations in the primitive variables formulation. The Reynolds number based on the cylinder diameter and free-stream speed of the flow is 200. The non-dimensional rotation rate, α (ratio of the surface speed and freestream speed), is varied between 0 and 5. The time integration of the flow equations is carried out for very large dimensionless time. Vortex shedding is observed for α < 1.91. For higher rotation rates the flow achieves a steady state except for 4.34 < α < 4:70 where the flow is unstable again. In the second region of instability, only one-sided vortex shedding takes place. To ascertain the instability of flow as a function of α a stabilized finite element formulation is proposed to carry out a global, non-parallel stability analysis of the two-dimensional steady-state flow for small disturbances. The formulation and its implementation are validated by predicting the Hopf bifurcation for flow past a non-rotating cylinder. The results from the stability analysis for the rotating cylinder are in very good agreement with those from direct numerical simulations. For large rotation rates, very large lift coefficients can be obtained via the Magnus effect. However, the power requirement for rotating the cylinder increases rapidly with rotation rate.

Instabilities and spatio-temporal chaos of long-wave hexagon patterns in rotating Marangoni convection.

We consider surface-tension driven convection in a rotating fluid layer. For nearly insulating boundary conditions we derive a long-wave equation for the convection planform. Using a Galerkin method and direct numerical simulations we study the stability of the steady hexagonal patterns with respect to general side band instabilities. In the presence of rotation, steady and oscillatory instabilities are identified. One of them leads to stable, homogeneously oscillating hexagons. For sufficiently large rotation rates the stability balloon closes, rendering all steady hexagons unstable and leading to spatio-temporal chaos. (c) 2002 American Institute of Physics.

Low-Prandtl-number convection in a rotating cylindrical annulus

Motivated by recent experimental results obtained in a low-Prandtl-number fluid (Jaletzky 1999), we study theoretically the rotating cylindrical annulus model with rigid boundary conditions. A boundary layer theory is presented which allows a systematic study of the linear properties of the system in the asymptotic regime of very fast rotation rates. It shows that the Stewartson layers have a (de)stabilizing influence at (high) low Prandtl numbers. In the weakly nonlinear regime and for low Prandtl numbers, a strong retrograde mean flow develops at quadratic order. The Poiseuille part of this mean flow is determined by an equation obtained by averaging of the Navier–Stokes equation. It thus gives rise to a new global-coupling term in the envelope equation describing modulated waves, which can be used for other systems. The influence of this global-coupling term on the sideband instabilities of the waves is studied. In the strongly nonlinear regime, the waves restabilize against these instabilities at small rotation rates, but they are destabilized by a short-wavelength mode at larger rotation rates. We also find an inversion in the dependence of the amplitude on the Rayleigh number at low Prandtl numbers and intermediate rotation rates.

Linear Stability of Thermal Convection in Rotating Systems with Fixed Heat Flux Boundaries

Linear stability of rotating thermal convection in a horizontal layer of Boussinesq fluid under the fixed heat flux boundary condition is examined by the use of a vertically truncated system up to wavenumber one. When the rotation axis is in the vertical direction, the asymptotic behavior of the critical convection for large rotation rates is almost the same as that under the fixed temperature boundary condition. However, when the rotation axis is horizontal and the lateral boundaries are inclined, the mode with zero horizontal wavenumber remains as the critical mode regardless of the rotation rate. The neutral curve has another local minimum at a nonzero horizontal wavenumber, whose asymptotic behavior coincides with the critical mode under the fixed temperature condition. The difference of the critical horizontal wavenumber between those two geometries is qualitatively understood by the difference of wave characteristics; inertial waves and Rossby waves, respectively.

Instabilities of magnetically induced shear layers and jets

We investigate numerically the flow of an electrically conducting fluid confined in a spherical shell, with the inner sphere rotating, the outer sphere stationary, and a strong magnetic field imposed parallel to the axis of rotation. It has previously been shown that the axisymmetric basic state depends strongly on the electromagnetic boundary conditions used, with insulating boundaries yielding a shear layer, but conducting boundaries yielding a counter–rotating jet, where in both cases these structures are located on the cylinder parallel to the imposed field and tangent to the inner sphere. Here we compute the non–axisymmetric instabilities of these basic states, and show that for sufficiently large rotation rates both the shear layer and the jet spawn a series of vortices encircling the tangent cylinder. Finally, we consider the fully three–dimensional nonlinear equilibration, and show that in the supercritical regime a secondary bifurcation occurs in which the number of vortices (for the shear layer) or vortex pairs (for the jet) is reduced by one.

Equation of state of dense nuclear matter and an upper bound on neutron star masses

We have discussed, in general, the important physical parameters, likemaximum mass, radius, and the minimum rotation period of self-bound,causally consistent, and pulsationally stable neutron stars (Q-starmodels) by using a realistic stiff EOS (such that, the speed of sound,v∝ PN, or nP=K(E-Ea)n, where K≤ 1 and n =1/(1-2N);where P and E represent respectively, the pressure and theenergy-density, and Ea is the value of E at the surface (r = a) of the configuration) within the two constraints imposed by: (i) The minimumrotation period, Prot, for the pulsar known to date corresponds to1.558 ms, and (ii) The maximum number density anywhere inside thestructure for the models described as Q-stars cannot exceed ∼ 1nucleon/fm3. By using the empirical formula given by Koranda,Stergioulas and Friedman (1997) (KSF-formula), and imposing constraint(i), we have obtained an upper bound of Mmax ≅ 7.76 M⊙radius a≅ 32.5 km, and the central energy-density around 2.17 ×1014 g cm-3 (for n =1.01). Constraint (ii) provides the minimumrotation period, Prot ∼ 0.489 ms for the maximum mass Mmax∼ 2.4 M⊙, and the central energy-density around 2.20 ×1015 g cm-3 (for n =1.01). The speed of sound at the centre ofthese models approaches ∼ 99% of the speed of light `c' (in thevacuum) and vanishes at the surface of the configuration together withpressure. If we relax the maximum Kepler frequency imposed by the fastestrotating pulsar known to date (constraint (i)), in view of certainobservational effects and theoretical evidences, and allow the present EOSto produce larger rotation rates than the 1.558 ms pulsar, the maximummass of the non-rotating model drops down to a value ∼ 7.2 M⊙.The higher values of masses (≥ 7 M⊙) and radii (∼31-32 km) obtained in this study imply that these models may representthe massive compact objects like Cyg X-1, Cyg XR-1, LMC X-3, and otherswhich are known as black hole candidates (BHCs). This study also suggestthat the strongest contender for black hole at present might be recurrentnova V404 Cyg (mass estimate ∼ 8 -12 M⊙).

Numerical study of tilt stability of prolate field-reversed configurations

Global stability of the field-reversed configuration (FRC) has been investigated numerically using both three-dimensional magnetohydrodynamic and hybrid (fluid electron and δf particle ion) simulations. The stabilizing effects of velocity shear and finite ion Larmor radius (FLR) on the n=1 internal tilt mode in the prolate FRCs have been studied. Sheared rotation is found to reduce the growth rate, however a large rotation rate with Mach number of M≳1 is required in order for significant reduction in the instability growth rate to occur. Kinetic effects associated with large thermal ion orbits have been studied for different kinetic equilibria. The simulations show that there is a reduction in the tilt mode growth rate due to FLR effects, but complete linear stability has not been found, even when the thermal ion gyroradius is comparable to the distance between the field null and the separatrix. The instability existing beyond the FLR theory threshold could be due to the resonant interaction of the wave with ions whose Doppler shifted frequency matches the betatron frequency.

An analytical model of velocity-derivative skewness of rotating homogeneous turbulence

The velocity-derivative skewness of rotating homogeneous turbulence is analytically derived from the turbulent kinetic energy dissipation rate equation by assuming a simple energy spectrum function. It predicts the cut-off of spectral energy at large rotation rate, and the predicted reduction of skewness due to rotation agrees well with the results of direct numerical simulation and eddy-damped quasi-normal Markovian closure.

Transition to Temporal Chaos in anO(2)-Symmetric Convective System for Low Prandtl Numbers

Two-dimensional nonlinear convection in a vertical rotating cylindrical annulus with flat adiabatic stress-free lids, heated from the inside and with radial gravity, is numerically analyzed for a low value of the Prandtl number, σ = 0.025. When the Rayleigh number exceeds a critical value, the conduction state becomes unstable and steady columns parallel to the axis of rotation, and characterized by a finite integer azimuthal wavenumber, n, are the preferred form of convection at the onset for large rotation rates. Despite the presence of rotation, equations retain the O(2) symmetry for z-independent columnar solutions. Both by using continuation techniques and by a time-integration of the evolution equations, primary nonlinear solutions are obtained for a moderate value of the radius ratio, and are found to give way to periodic solutions in the form of direction reversing traveling waves. The new solution keeps the same wavenumber and breaks the reflection symmetry of the columns. As a consequence, an oscillatory mean flux appears that decreases the efficiency of the heat transport in the radial direction. By further increasing the Rayleigh number, a transition from the oscillatory to a chaotic flow takes place. This chaotic state is reached via a pitchfork bifurcation that breaks the rotation symmetry R2π/3 of the orbit, followed by a subcritical Neimark-Sacker bifurcation that gives rise to a quasi-periodic solution. Finally, the invariant torus breaks up and a chaotic regime appears.

Mean zonal flows excited by critical thermal convection in rotating spherical shells

Mean zonal flows excited by critical modes of Boussinesq convection in rotating spherical shells are evaluated systematically by the weak nonlinear method. Calculations are executed in the ranges of the Prandtl number 0.01 ≤ P ≤ 102, the Taylor number 100 ≤ T ≤ 107, and the radius ratio 0.2 ≤ η ≤ 0.8. In the case of small rotation rates, two new states of zonal flow at the equatorial outer boundary are obtained; westerly flow emerges even when the Prandtl number is large, and easterly flow emerges even when the Prandtl number is small. The latter case is interesting since the induced equatorial easterly is in the opposite direction to that of the acceleration. In the case of large rotation rates, it is confirmed that the direction of mean zonal flow at the equatorial outer boundary does not depend on the thickness of the shell.

A note on stabilising and destabilising effects of Ekman boundary layers

Marginal instability of a Bénard layer is considered in the asymptotic case of large rotation rate, i.e., for E → 0, where E = v/ωd 2 is the Ekman number, v is the kinematic viscosity, ω is the angular velocity and d is the depth of the layer. The nature of the convection is determined by the magnitude of the Prandtl number Pr = v/κ, where κ is the thermal diffusivity. The cases Pr > O(E) are studied here; the remaining possibility of thermoinertial waves arising when Pr > O(E) has recently been analysed by us elsewhere (Phys. Fluids 9, 1980–1987, 1997). Attention is focused here on the role of the Ekman boundary layer in determining the critical value, Rc , of the Rayleigh number, R, at which convection is marginally possible, where R = gαβd2 /ωκ, g is the gravitational acceleration, β is the applied temperature gradient and α is the thermal expansion coefficient. For Pr ⩾ O(1), the marginal state is steady convection and where k = 0 when both the bounding surfaces are stress-free, k = 1 when one surface is stress-free and the other is non-slip, and k = 2 when both the bounding surfaces are non-slip. For O(E) < Pr < O(1), the marginal state is oscillatory convection and The second terms in these expressions for Rc , representing Ekman layer corrections to the leading order E −1/3 terms, can be substantial because of the small exponent of E. In steady convection, viscous dissipation in the Ekman layers destabilises the Benard layer, but in oscillatory convection it is stabilising.

Spatiotemporal Patterns in Liquid-Liquid Taylor-Couette-Poiseuille Flow

The vortex structure of immiscible liquid-liquid Taylor-Couette-Poiseuille flow was studied using photographic techniques. Several parameters were considered, including the feed composition and the inner cylinder rotation rate. For certain feed compositions and sufficiently large rotation rates a translating banded structure, which consisted of alternating aqueous and organic-rich vortices, persisted indefinitely. At lower rotation rates, either a spatially homogeneous emulsion evolved or sustained oscillations between the banded and homogeneous structures developed.

Transverse motion of a disk through a rotating viscous fluid

A thin rigid disk translates edgewise perpendicular to the rotation axis of an unbounded fluid undergoing solid-body rotation with angular velocity Ω. The disk face, with radius a, is perpendicular to the rotation axis. For arbitrary values of the Taylor number, [Tscr ] = Ωa2/ν, and in the limit of zero Reynolds number [Rscr ]e, the linearized viscous equations reduce to a complex-valued set of dual integral equations. The solution of these dual equations yields an exact representation for the velocity and pressure fields generated by the translating disk.For large rotation rates [Tscr ] [Gt ] 1, the O(1) disturbance velocity field is confined to a thin O([Tscr ]−1/2) boundary layer adjacent to the disk. Within this boundary layer, the flow field near the disk centre undergoes an Ekman spiral similar to that created by a nearly geostrophic flow adjacent to an infinite rigid plate. Additionally, flow within the boundary layer drives a weak O([Tscr ]−1/2) secondary flow which extends parallel to the rotation axis and into the far field. This flow consists of two counter-rotating columnar eddies, centred over the edge of the disk, which create a net in-plane flow at an angle of 45° to the translation direction of the disk. Fluid is transported axially toward/away from the disk within the core of these eddies. The hydrodynamic force (drag and lift) varies as O([Tscr ]1/2) for [Tscr ] [Gt ] 1; this scaling is consistent with the viscous stresses created in the Ekman boundary layer. Additionally, an approximate expression, suitable for all Taylor numbers, is given for the hydrodynamic force on a disk translating broadside along the rotation axis and edgewise transverse to the rotation axis.

Winds from rotating Wolf-Rayet stars: the wind-compressed zone model

Theoretical models for Wolf-Rayet winds that include the effects of rotation and magnetic fields are discussed. These are of two types: magnetic rotator models and wind compression models. The magnetic rotator models are examples of equatorial expulsion models, typically requiring large rotation rates to produce significant equatorial density enhancements. Wind- compression models are introduced here as an application of the two-dimensional Wind-Compressed Disk (WCD) model that has recently been developed for Be stars. An equatorial disk forms because of the supersonic confluence of the flow from the upper and lower hemispheres of the star. In the application to WR stars, we suggest that disk formation is not required, and instead a less extreme example that we call a wind-compressed zone (WCZ) may suffice. Because the winds of WR stars have a geometrically extended acceleration region, or a “slow” velocity law, there can be compressions by an order of magnitude, even with moderate stellar rotation rates of about 16% critical. Magnetic flux conservation is then used to estimate the enhancement of the field strength that occurs because of the equatorial wind-compression. We consider an application of the model to explain the occurrence of polarization and dust formation among only some WR stars. Finally, we consider the WR + compact companion system WR 147, and suggest that the enhanced accretion in a WCZ could help explain the large X-ray emission.

Tomography on tokamak fusion test reactor

Reconstruction techniques have been developed to obtain 2D images of plasmas from the electron cyclotron emission (ECE) and soft x-ray emission signals by taking advantage of the large toroidal rotation rate on the tokamak fusion test reactor (TFTR). From the numerical examination using the sawtooth crash model, we found that the rotational tomography is effective if the crash time is longer than the two rotation periods.

Asymptotic theory of wall-attached convection in a rotating fluid layer

Asymptotic expressions for the onset of convection in a horizontal fluid layer of finite extent heated from below and rotating about a vertical axis are derived in the limit of large rotation rates in the case of stress-free upper and lower boundaries. In the presence of vertical sidewalls, the critical Rayleigh number Rc is much lower than the classical value for an infinitely extended layer. In particular, we find that Rc grows in proportion to τ when the sidewall is insulating, where τ is the dimensionless rotation rate. When the sidewall is infinitely conducting, Rc grows in proportion to as in the case of an infinitely extended layer but with a lower coefficient of proportionality. Numerical results obtained at finite values of τ show good agreement with the asymptotic formulae.

Barotropic flow over bottom topography— experiments and nonlinear theory

Abstract Barotropic flow over finite amplitude two-wave bottom topography is investigated both experimentally and theoretically over a broad parameter range. In the experiments, the fluid is contained in a vertically oriented, rotating circular cylindrical annulus. It is forced into motion relative to the annulus by a differentially rotating, rigid, radially sloping lid in contact with the top surface of the fluid. The radial depth variation associated with the slope of the lid, and an equal and opposite slope of the bottom boundary, simulates the effect of the variation of the Coriolis parameter with latitude (β) in planetary atmospheres and in the ocean. The dimensionless parameters which control the fluid behavior are the Rossby number (ϵ), the Ekman number (E), the β parameter, the aspect ratio (δ), the ratio of the mean radius to the gap width (α) and the ratio of the topographic height to the mean fluid depth (η). The Rossby and Ekman numbers are varied over an order of magnitude by conducting experiments at different rotation rates of the annulus. Velocity measurements using photographs of tracer particles suspended in the fluid reveal the existence of a stationary, topographically forced wave superimposed on an azimuthal mean current. With successively larger rotation rates (i.e. lower ϵ and E) the wave amplitude increases and then levels off, the phase displacement of the wave upstream of the topography increases and the azimuthal mean velocity decreases and then levels off. Linear quasigeostophic theory accounts qualitatively, but not quantitatively, for the phase displacement, predicts the wave amplitude poorly and provides no basis for predicting the zonal mean velocity. Accordingly, we have solved the nonlinear, steady-state, quasigeostrophic barotrophic vorticity equation with both Ekman layer and internal dissipation using a spectral colocation method with Fourier representation in the azimuthal direction and Chebyshev polynomial representation in the radial direction. For boundary conditions at the side walls, we specified zero velocity. Side wall boundary layers then appear explicitly in the numerical solution. At the bottom and top of the fluid, we specified that the vertical velocity at the mean height of each boundary is the sum of two components—one forced by Ekman suction in the absence of topography and the other by the condition that there can be no flow normal to the rigid boundary. We justify this choice by the smallness of the Ekman number and of the radial and azimuthal slopes of the topography. We have found that the use of three Fourier components and seven Chebyshev polynomials is sufficient to account qualitatively for the experimental results, although small quantitative discrepancies suggest that further investigation of the neglect of effects originally considered to be small is needed.

The configurations of a FENE bead-spring chain in transient rheological flows and in a turbulent flow

The changes in the configuration of a FENE bead-spring chain in a direct numerical simulation of turbulent channel flow and in some simple rheological flows are examined. Unraveling occurs both in uniaxial and shear flows, but the uniaxial flow is more effective. A vortex with a large rotation rate perpendicular to the principal strain of a uniaxial flow has only a minor retarding effect while a small rotation rate delays the unraveling substantially. In a turbulent flow, the chain unravels the most in the viscous sublayer, to about 90% of its fully extended length. It aligns at a 7° angle with the direction of mean flow. In the buffer zone, it unravels and coils up and takes different orientations at different times. Outside the wall region, the chain assumes a coiled configuration. The unraveling of the chain strongly depends on the relaxation time of the chain normalized with the wall shear rate, λ+. A value of λ+=10 exhibits strong unraveling while very weak unraveling is observed below λ+=1.

Stabilization and destabilization of turbulent shear flow in a rotating fluid

We consider turbulent shear flows in a rotating fluid, with the rotation axis parallel or antiparallel to the mean flow vorticity. It is already known that rotation such that the shear becomes cyclonic is stabilizing (with reference to the non-rotating case), whereas the opposite rotation is destabilizing for low rotation rates and restabilizing for higher. The arguments leading to and quantifying these statement are heuristic. Their status and limitations require clarification. Also, it is useful to formulate them in ways that permit direct comparison of the underlying concepts with experimental data.An extension of a displaced particle analysis, given by Tritton &amp; Davies (1981) indicates changes with the rotation rate of the orientation of the motion directly generated by the shear/Coriolis instability occurring in the destabilized range.The ‘simplified Reynolds stress equations scheme’, proposed by Johnston, Halleen &amp; Lezius (1972), has been reformulated in terms of two angles, representing the orientation of the principal axes of the Reynolds stress tensor (αa) and the orientation of the Reynolds stress generating processes (αb), that are approximately equal according to the scheme. The scheme necessarily fails at large rotation rates because of internal inconsistency, additional to the fact that it is inapplicable to two-dimensional turbulence. However, it has a wide range of potential applicability, which may be tested with experimental data. αaand αbhave been evaluated from numerical data for homogeneous shear flow (Bertoglio 1982) and laboratory data for a wake (Witt &amp; Joubert 1985) and a free shear layer (Bidokhti &amp; Tritton 1992). The trends with varying rotation rate are notably similar for the three cases. There is a significant range of near equality of αaand αb. An extension of the scheme, allowing for evolution of the flow, relates to the observation of energy transfer from the turbulence to the mean flow.

Partitioning of non-coaxiality in deforming layered rock masses

Partitioning of non-coaxiality and other flow parameters in deforming layered rock masses with nearly incompressible Newtonian rheology is examined in two dimensions by the finite element method. The results show that layers with viscosity contrast in suitable orientation deform by different and even by a reverse sense of non-coaxiality. This remarkable partitioning of non-coaxiality is due to a large rotation rate of instantaneous stretching axes in a heterogeneous body as a result of geometry changing through deformation rather than through rotation of material lines.